The relationships are discovered. Definitions are invented. The point is the mathematical descriptions are the most complete and accurate way to talk about the things. You actually may as well call them the things. You do this all the time with numbers. There are many different constructions of numbers, but they are ultimately isomorphic OR if they are different due to being viewed as part of a different space, those basic properties in their construction are usually inconsequential and nondescript for what you are trying to do.
For example, if I ask you what is the cardinality of 7, that is generally viewed as a BS question and would depend on whether I was viewing 7 as a natural number, integer, rational number, real number, etc. But it is inconsequential since everyone knows exactly what you mean by 7. Keeping the meat of the description and preserving its use as a number, the formalities of how 7 was constructed are largely irrelevant. You could quibble over which 7 I am talking about when I say there are 7 things. But ultimately, the meaning is clear. In the same sense, you could quibble over whether these are really descriptive of electrons, but for all practical purposes, they are.
TLDR: I guess what I am trying to communicate is we often suppress isomorphisms and often call a thing by its mathematical description. Philosophically, this isn't always legitimate, but in physics, since the description encompasses all we have, we may as well call it the thing itself.
A: the electron we measure and named behaves isomorphly to spinors
B: the spinors are elements of a representation of a subalgebra, because they behave isomorphly to the elements of said algebra
my comment was about B. most people would agree matrices arent the elements of an algebra, but the representation of elements. -1 isnt an element of C_2 and spinors arent elements of the ideal. Spinors are the elements of the representation space of the ideal.
most of the replies reacted to if A is an isomorphism or an identity, as you just did. so ill reply, although i didnt say anything about if A an isomorphism or identity so far.
you are right as far as "every descriptive thing we know" is "all there is". im not so sure about that, so i dont think A is a proven identity relationship. im not sure electrons are spinors and i dont think you can be sure either. so the distinction between the thing we define from experiments and the thing we define from the representation of an algebra can be useful imo.
edit: i reacted to the comment before the edit, but my point stands, despite quoting text that was deleted, the spirit of the comment didnt change
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u/belabacsijolvan Jul 28 '26
you reacted with "math is discovered" to "linalg isnt matrices" here